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When considering the grand potential for a photon gas, one encounters an integral of the form:

[tex] \Sigma = a\int_{0}^{\infty}x^2\ln\Big(1 - e^{-bx}\Big)dx [/tex]

I have never had to integrate something like this before; I was told it is done via contour integration, but I have never used such a method and the examples typically given are not of this form. Could somebody please provide some assistance? I have tried to learn a bit myself, but I remain perplexed. What would be the contour enclosing such an integral, for example? Thanks.

UPDATE: I noticed that integration by parts puts the integral in a form of [tex]\int_{0}^{\infty}\frac{x^3dx}{e^{bx} - 1}[/tex] ignoring constants. I now recognize this as a familiar integral found in Stefan's Law, but it would still be nice to see someone's perspective of the contour method to evaluate it.

[tex] \Sigma = a\int_{0}^{\infty}x^2\ln\Big(1 - e^{-bx}\Big)dx [/tex]

I have never had to integrate something like this before; I was told it is done via contour integration, but I have never used such a method and the examples typically given are not of this form. Could somebody please provide some assistance? I have tried to learn a bit myself, but I remain perplexed. What would be the contour enclosing such an integral, for example? Thanks.

UPDATE: I noticed that integration by parts puts the integral in a form of [tex]\int_{0}^{\infty}\frac{x^3dx}{e^{bx} - 1}[/tex] ignoring constants. I now recognize this as a familiar integral found in Stefan's Law, but it would still be nice to see someone's perspective of the contour method to evaluate it.

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